Does the term "monad" apply to values of types like Maybe or List, or does it instead apply only to the types themselves?

Question

I've noticed that the word "monad" seems to be used in a somewhat inconsistent way. I've come to believe that this is because many (if not most) of the monad tutorials out there are written by folks who have only just started to figure monads out themselves (eg: nuclear waste spacesuit burritos), and so the term ends up getting kind of overloaded/corrupted.

In particular, I'm wondering whether the term "monad" can be applied to individual values of types like Maybe, List or IO, or if the term "monad" should really only be applied to the types themselves.

This is a subtle distinction, so perhaps an analogy might make it more clear. In mathematics we have, rings, fields, groups, etc. These terms apply to an entire set of values along with the operations that can be performed on them, rather than to individual elements. For example, integers (along with the operations of addition, negation and multiplication) form a ring. You could say "Integer is a ring", but you would never say "5 is a ring".

So, can you say "Just 5 is a monad", or would that be as wrong as saying "5 is a ring"? I don't know category theory, but I'm under the impression that it really only makes sense to say "Maybe is a monad" and not "Just 5 is a monad".

Answer 1

"Monad" (and "Functor") are popularly misused as describing values. No value is a monad, functor, monoid, applicative functor, etc. Only types & type constructors (higher-kinded types) can be. When you hear (and you will) that "lists are monoids" or "functions are monads", etc, or "this function takes a monad as an argument", don't believe it. Ask the speaker "How can any value be a monoid (or monad or ...), considering that Haskells classes classify types (including higher-order ones) rather than values?" Lists are not monoids (etc). List a is.

My guess is that this popular misuse stems from mainstream languages having value classes and not type classes, so that habitual, unconscious value-class thinking sneaks in.

Why does it matter whether we use language precisely? Because we think in language and we build & convey understandings via language. So in order to have clear thoughts, it helps to have clear language (or be able to at any time).

"The slovenliness of our language makes it easier for us to have foolish thoughts. The point is that the process is reversible." - George Orwell, Politics and the English Language

Edit: These remarks apply to Haskell, not to the more general setting of category theory.